One-loop Corrected Holographic Shear Viscosity to Entropy Density Ratio at Low Temperatures

Abstract

Near-extremal black holes contain infrared-enhanced quantum fluctuations in their near-horizon near-AdS${}_2$ throat, governed in part by Schwarzian modes. We study the effect of these fluctuations on the zero-frequency stress-tensor response of a near-extremal asymptotically AdS${}_4$ Reissner--Nordström black brane whose transverse directions are regulated by a finite toroidal quotient. Working directly in four dimensions, we compute the leading correction in the weakly coupled Schwarzian regime $T_q\ll T\ll r_0/L^2$. At one loop, the shear perturbation couples to the $n=2$ would-be zero mode and produces a correction proportional to $T_q/T$. Together with the logarithmic correction to the entropy, this yields a finite-volume correction to $\eta /s$. Our calculation provides a direct four-dimensional benchmark, complementary to exact two-dimensional treatments, while keeping the asymptotic AdS${}_4$ source explicit.

Revised version. Clarified the regime of validity